The Invariance Approach to Structure and Dynamics: Classical Hyperspherical Coordinates

The hyperspherical coordinate systems have been extensively adopted in the study of few-body scattering problems of nuclear and molecular physics, for example in the practical implementation of demanding quantum calculations typical of chemical reactions. Hyperangular momenta are the dynamical quantities used in this representation and the hyperspherical harmonics are the corresponding basis functions. The use of such formalism is limited to the treatment of three- or four-center problems, due to the exceedingly high computational cost of quantum dynamics calculations. To circumvent this restriction, an hyperspherical formulation has been developed in a series of works during the last decades, suitable for the simulation of cluster and large molecular system dynamics. Such a hyperspherical formulation is based on classical definitions of the hyperangular momenta and on the search of invariant dynamical quantities. Exploiting invariance of hyperspherical shape coordinates with respect to rotations and kinematic rotations, we consider the use of the classical hyperspherical representation as a tool for the static analysis of geometry and minimum energy structures of atomic and molecular structures, showing some applications to relatively simple systems.